Show that all harmonics are present on a stretched string between two rigid supports.

Explanation

Consider a string of length `l`, stretched under tension `T` between two fixed points. If the string is plucked and then released, transverse harmonic wave propagates along its length and is reflected at the end.

Frequency of vibrations is given by `n=(1)/(lambda)sqrt((T)/(m))`
For obtaining `P` loops in string , it has to be plucked at a distance `l//2P` from one fixed end.
For first mode of vibration:
`lambda_(1)=2l`
`l=(lambda_(1))/(2)`
If `n_(1)` be the frequency of vibration of the string and `v` the speed of wave in the string
`n_(1)=(v)/(lambda_(1))=(v)/(2l)`
`n_(1)=(1)/(2l)xxsqrt((T)/(m))`..........`(i)`
This is the fundamental frequency for first harmonic. For second mode of vibrations:
`lambda_(2)=(2l)/(2)`
`:.n_(2)=(2)/(2l)xxsqrt((T)/(m))`
`n_(2)=2n_(1)` [from equation `(i)`]
For third mode of vibrations :
`lambda_(3)=(2l)/(3)`
`n_(3)=(3)/(2l)xxsqrt((T)/(m))=3n_(1)` [from equation `(i)`]
In general, if the string is plucked at length `l//2P`, then it vibrates in `P` segments (loops) and we have `P^(th)` harmonic
`n_(P)=(P)/(2l)sqrt((T)/(m))`
Thus, the frequencies of the fundamental tone and the overtones of a stretched string have the following relationship :
`n_(1): n_(2) : n_(3).........., n_(P)=1 : 2 :3 ...... : p`
These frequencies are in harmonic series with both even and odd harmonics.